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Tuning discrete PI controllers for direction-dependent systems with delay
2nd IFAC Workshop on Linear Parameter Varying Systems 2nd IFAC Workshop on Linear Parameter Varying Systems Florianopolis, Brazil,on September 3-5, 2018Varying Systems 2nd IFAC Workshop Linear Parameter Florianopolis, Brazil, September 3-5, 2018 Available online at www.sciencedirect.com 2nd IFAC Workshop Linear Parameter Florianopolis, Brazil,on September 3-5, 2018Varying Systems Florianopolis, Brazil, September 3-5, 2018
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IFAC PapersOnLine 51-26 (2018) 124–129
Tuning discrete PI controllers Tuning discrete PI controllers Tuning discrete PI controllers direction-dependent systems with Tuning discrete PI controllers direction-dependent systems with direction-dependent systems direction-dependent systems with with J. Toledo ∗∗ D. Sbarbaro ∗∗
for for for delay for delay delay delay
J. Toledo ∗∗∗ D. Sbarbaro ∗∗∗ J. Toledo ∗ D. Sbarbaro ∗ ∗ J. Toledo D. Sbarbaro of Electrical o ∗ ∗ Department of Electrical Engineering, Engineering, Universidad Universidad de de Concepci´ Concepci´ on, n, ∗ Department ∗ Concepci´ o n, Chile Engineering, (e-mail: [email protected], Department of Electrical Universidad de Concepci´ on, ∗ Concepci´ o n, Chile (e-mail: [email protected], Department of Electrical Engineering, Universidad de Concepci´ on, [email protected]) Concepci´ on, Chile (e-mail: [email protected], [email protected]) Concepci´ on, Chile (e-mail: [email protected], [email protected]) [email protected]) Abstract: Abstract: Direction-dependent Direction-dependent models models can can represent represent aa wide wide range range of of nonlinear nonlinear systems. systems. The The design of a Direction-dependent discrete time PI controller for these type ofa systems withoftime delay systems. is addressed. Abstract: models can represent wide range nonlinear The design of a discrete time PI controller for these type of systems with time delay is addressed. Abstract: models can represent wide range oftime nonlinear systems. The The asymptotic stability of the closed loop system is established by means ofdelay Lyapunov theory. design of a Direction-dependent discrete time of PIthe controller for these type ofa systems with is addressed. The asymptotic stability closed loop system is established by means Lyapunov theory. design of a discrete time PIthe controller for these type of systems with timeof delay is addressed. tuning strategy is based on solving an optimization problem with LMI constraints. A The asymptotic stability of closed loop system is established by means of Lyapunov theory. The asymptotic tuning strategy is based on solving an optimization problem with of LMI constraints. A The stability of the the on closed loop an system isproposed established by means Lyapunov theory. simple example illustrates effectiveness of the approach. Extensions to include The tuning strategy is based solving optimization problem with LMI constraints. A simple example illustrates the effectiveness of the proposed approach. Extensions to include The strategy is based solving anofoptimization problem withExtensions LMI are constraints. A more tuning complex temporal closed-loop specifications and switched PI controllers part of our simple example illustrates the on effectiveness theand proposed approach. to include more complex temporal closed-loop specifications switched PI controllers are part of our simple example illustrates the effectiveness of the proposed approach. Extensions to include current interest. more complex closed-loop specifications and switched PI controllers are part of our current interest.temporal more complex current interest.temporal closed-loop specifications and switched PI controllers are part of our © 2018, IFAC (International Federation of Automatic Control) Hosting by ElsevierInequality. Ltd. All rights reserved. current interest. Keywords: Keywords: PID PID controller, controller, direction-dependent direction-dependent dynamics, dynamics, Linear Linear Matrix Matrix Inequality. Keywords: PID controller, direction-dependent dynamics, Linear Matrix Inequality. Keywords: PID controller, direction-dependent dynamics, Linear Matrix Inequality. 1. INTRODUCTION ponential stability of the closed loop system are obtained. obtained. 1. INTRODUCTION ponential stability of the closed be loop systemtoare These results can directly applied the class 1. INTRODUCTION ponential stability of the closed be loop systemtoare These results can not not directly applied theobtained. class of of 1. INTRODUCTION ponential stability of the closed loop system are obtained. systems addressed in this work, since the system time delay Direction-dependent systems are non linear system such These results can not directly be applied to the class of addressed in this work, since the system time delay Direction-dependent systems are non linear system such systems results can in not directly betoapplied tothis the class of is direction dependent. In order address problem, that their dynamic switches different Direction-dependent systemsbetween are nontwo linear systemmodes such These systems addressed this work, since the system time delay is direction dependent. Inwork, ordersince to address thistime problem, that their dynamic switches between two different modes systems addressed in this the system delay Direction-dependent systems are non linear system such in this work, a simple transformation is proposed in order depending on the direction of the input or outputmodes vari- in direction In order to address this problem, that their dynamic switches of between two different this work,dependent. a simple transformation is proposed in order depending on theofdirection the input or outputmodes vari- is is direction InLyapunov-Krasovskii order to address thiscandidate problem, that switches can between two different to define aa dependent. CLKF. ables.their Thisdynamic kind systems be input found in many indusin this work, a simpleThe transformation is proposed in order depending on the direction of the or output varito define CLKF. The Lyapunov-Krasovskii candidate ables. This kind of systems can be found in many indusin this work, a simple transformation is proposed in et order depending on the direction of the input or output vari- function isa similar to theLyapunov-Krasovskii one proposed in [Chen al. trial processes such as sedimentation, heating or cooling, to define CLKF. The candidate ables. This kind of systems can be found in many industrial processes such as sedimentation, heating or cooling, function is similar to the one proposed in [Chen et al. to define a CLKF. The Lyapunov-Krasovskii candidate ables. This kind of systems can be found in many indus2003] and [Chen et al. 2005]. chemical, among others. Tan [2009] provides a unified view function is similar to the one proposed in [Chen et al. trial processes such as sedimentation, heating or cooling, 2003] and [Chen et al. 2005]. chemical, among others. Tan [2009] provides a unified view is[Chen similar to 2005]. the one proposed in [Chen et al. trial processes such as systems. sedimentation, heating or cooling, of direction-dependent These type ofa unified systems can function 2003] and et al. chemical, among others. Tan [2009] provides view is as of direction-dependent systems. These type of systems can The 2003]paper and [Chen et al. 2005]. chemical, among Tan provides view paper is organized organized as follow. follow. Sections Sections 22 describes describes be direction-dependent regarded as a others. special type[2009] ofThese linear switching systems; of systems. type ofa unified systems can The the characteristics of the direction-dependent In be regarded as a special type of linear switching systems; The paper is organized as follow. Sections 2systems. describes of direction-dependent systems. These type of systems can the characteristics of the direction-dependent systems. In where the switching function is defined by the process be regarded as a special type of linear switching systems; The paper is organized asmodels follow.are Sections 2systems. describes where the switching function islinear defined by thesystems; process section 3, the closed-loop described and the the characteristics of the direction-dependent In be regarded as a special type of switching 3, the closed-loop models are described and the itself. the switching function is defined by the process section where characteristics offor theclosed-loop direction-dependent systems. In necessary stability are obtained. section 3, conditions the closed-loop models are described and the itself. the switching function is defined by the process the where necessary conditions for closed-loop stability are obtained. itself. section 3,describes the closed-loop models to are described and the Section 4 a methodology tune a PI controller. The stability of linear switching system not only depends necessary conditions for closed-loop stability are obtained. itself. The stability of linear switching system not only depends Section a for methodology tostability tune a PI controller. necessary conditions closed-loop are obtained. Sections44 5describes illustrates the main results means of an on the estructure andswitching the parameters of only the different describes a methodology to tuneby a PI controller. The stability of linear system not depends Section Sections 5 illustrates the main results by means of an on the estructure and the parameters of the different Section 4 describes a methodology to tune a PI controller. The stability of linear switching system not only depends example. Finally in section 6 some conclusions and future modes, but also onand thethe switching function. Thedifferent neces- Sections 5 illustrates the main results by means of an on the estructure parameters of the example. Finally in section 6 some conclusions and future modes, but also on the switching function. The necesSections 5outlined. illustrates the main by means of an on estructure and the parameters of the different are sarythe andbut sufficient conditions for asymptotic stability of work example. Finally in section 6 someresults conclusions and future modes, also on the switching function. The neceswork are outlined. sary andbut sufficient for are asymptotic stability of example. Finally in section 6 some conclusions and future modes, also onconditions theswitching switching function. The necessystems with arbitrary surveyed in Lin and sary and sufficient conditions for asymptotic stability of work are outlined. systems with arbitrary switching are surveyed stability in Lin and sary andwith sufficient conditions asymptotic of work are 2. outlined. DIRECTION-DEPENDENT SYSTEMS Antsaklis [2009]. One ofswitching the keyfor assumption is that all the systems arbitrary are surveyed in Lin 2. DIRECTION-DEPENDENT SYSTEMS Antsaklis [2009]. One of the key assumption is that all and the systems with arbitrary switching are surveyed in Lin and 2. DIRECTION-DEPENDENT SYSTEMS modes are described by models of the same order. To the Antsaklis [2009]. Oneby of models the key of assumption that To all the the modes are described the same is order. 2. DIRECTION-DEPENDENT SYSTEMS Antsaklis [2009]. Oneby of the the key assumption is that To all the the In this work, best of our knowledge, study of direction-dependent modes are described models of the same order. In this work, we we will will consider consider SISO SISO systems systems with with directiondirectionbest of are ourdescribed knowledge, the study of direction-dependent modes by models of the same order. To the dependent dynamics which can be systems written as: systems with direction dependent time delays has not been In this work, we will consider SISO with directionbest of our knowledge, the study of direction-dependent dependent dynamics which can be written as: systems with direction dependent time delays has not been In this work, consider SISO systems with best of our knowledge, the study of direction-dependent addressed. The discrete models associated to these systems +be bσ(k) u(k −as: dσ(k)direction) x(kdynamics +we 1)will = A which can written systems with direction time delays has not been dependent σ(k) x(k) addressed. The discretedependent models modes associated to these systems bσ(k) u(k −as: dσ(k) x(kdynamics + 1) = Awhich dependent can+be written σ(k) (1) systems with direction dependent time delays has not been σ(k) x(k) σ(k) σ(k) ) lead to representations having described by models = (1) addressed. The discrete models associated to these systems x(k) + b u(k − d ) x(k + 1) A σ(k) σ(k) σ(k) y(k) = c x(k) σ(k) lead to representations described bysystems models addressed. Theorders. discrete having models modes associated to these y(k) x(k) + bσ(k) u(k − dσ(k) ) x(k + 1) = c Ax(k) (1) with different σ(k) σ(k) σ(k) lead to representations having modes described by models n = c x(k) with different orders. y(k) (1) x(k) ∈ R is the state vector, u(k) ∈ R is the input, n lead to representations having modes described by models where n= with different orders. where x(k) ∈ R is the state vector, u(k) ∈ R is the input, y(k) c x(k) n output and σ(k) is a discrete variable This work addresses the problem of tuning a PI controller where n y(k) ∈x(k) R is∈ the with different orders. R is the state vector, u(k) ∈ R is the input, This worktype addresses the problem of tuning a PIiscontroller y(k) ∈x(k) R is the andvector, σ(k)simplicity is a discrete variable where Rn isoutput the signal. state u(k) ∈ Rwe is the input, for of The analysis centered This worktype addresses the problem of tuning a PIiscontroller will write y(k) ∈ R is∈switching the output and For σ(k)simplicity is a discrete variable for these these of systems. systems. The stability stability analysis centered representing representing switching signal. For we will write n×n This work addresses the problem of tuning a PI controller y(k) ∈ R is the output and σ(k) is a discrete variable in finding a Common Quadratic Lyapunov Function for σ(k) for these type of systems. The stability analysis is centered as σ. The parameter of the plant are A ∈ R n×n representing switching signal. For simplicity weσσ will write n×n ,, σ(k) as σ. The parameter of the plant are A ∈ R in finding a Common Quadratic Lyapunov Function for n×1 1×n for these type of systems. The stability analysis is centered σ n×n representing switching signal. For simplicity we will write establishing the closed Quadratic loop stability [Leith Function et al. 2003]. n×n b ∈ R , c ∈ R and d ∈ N. Notice also that σ(k) as σ. The parameter of the plant are A ∈ R in finding a Common Lyapunov for n×1 1×n σ σ σ also n×n n×1 , c ∈ R1×n and d σ bσσ ∈asRσ. ∈ N. are Notice that,, establishing the closed loop stability [Leith et design al. 2003]. σ in finding a Common Quadratic Lyapunov Function for σ n×1 1×n σ(k) The parameter of the plant A ∈ R The use of a CQLF in the context of control for n×1 1×n σ also the time delay depends on σ(k). The switching signal, establishing the closed loop stability [Leith et al. 2003]. b ∈ R , c ∈ R and d ∈ N. Notice that σ σ σ timen×1 the delay depends σ(k). switching The use of athe CQLF in loop the context of[Leith control design for b establishing closed stability etby al. several 2003]. R (2), , ccan ∈ depend R1×n on and dσσ on ∈The N. Notice alsosignal, switched linear systems have been addressed σ ∈ The use of a CQLF in the context of control design for equation either the switching direction ofthat the the time delay depends on σ(k). The signal, switched linear systems have been addressed by several (2), can depend either onThe the switching direction of the The use For of ainstance, CQLF in the context of control design for equation the time delay depends on σ(k). signal, authors. Chaib et been al. [2006] provideby a method input: v(k) = u(k) − u(k − 1) or the output: v(k) = y(k) − equation (2), can depend either on the direction of the switched linear systems have addressed several v(k) = u(k) u(k − 1) or the v(k) = y(k) − authors. For instance, Chaib etfor al.continuous [2006] provide aswitched method switched linear systems have been addressed several input: equation (2), can − depend either onoutput: the direction of the to designFor dynamic controllers timeby y(k − 1). It takes its values over a finte set of values input: v(k) = u(k) − u(k − 1) or the output: v(k) = y(k) − authors. instance, Chaib et al. [2006] provide a method y(k − 1). It takes its values over a finte set of values to design dynamic controllers for continuous time switched authors. For instance, etfor al.continuous [2006] provide aswitched method v(k) =the u(k)different −its u(kvalues − 1) orover theFor output: v(k) = y(k) − linear system. Chen etChaib al. [2014] address the problem of input: representing modes. instance σ(k) = to design dynamic controllers time y(k − 1). It takes a finte set of values the different modes. For instance σ(k) = ii linear system. Chen etsignal al. [2014] address the problem of representing to design dynamic controllers for continuous time switched y(k − 1). It takes its values over a finte set of values designing a switching for exponential stabilization linear system. Chen et al. [2014] address the problem of means that the ith mode, represented by A , b and d , i isi representing theith different modes. For by instance designing a switching signal for exponential stabilization means that the mode, represented Aiii , biii σ(k) and d= ii , isi linear system. Chen etsignal al. system [2014] address thestabilization problem of representing theith different modes. For by instance σ(k) = of a discrete-time switched with time-varying delay. activated. means that the mode, represented A , b and d designing a switching for exponential ii ii ii , is activated. of a discrete-time switched system with time-varying delay. designing a switching signal for exponential stabilization means that the ith mode, represented by A , b and d Bya applying a Common Lyapunov-Krasovskii Functional i 0 i i , is 1 v(k) < activated. of discrete-time switched system with time-varying delay. By applying a Common Lyapunov-Krasovskii Functional 1 v(k) < 0 of a discrete-time switched system with time-varying delay. activated. (CLKF), sufficient conditions to guarantee the global ex2 v(k) > σ(k) (2) By applying a Common Lyapunov-Krasovskii v(k) (CLKF), sufficient conditions to guarantee theFunctional global exv(k) < > 0000 σ(k) = = 112 (2) By applying a Common Lyapunov-Krasovskii v(k) < (CLKF), sufficient conditions to15110019 guarantee theFunctional global exσ(k − 1) v(k) = 0 2 v(k) > 0 σ(k) = (2) The support of Project FONDAP is acknowledged σ(k − 1) v(k) = 0 The support (CLKF), sufficient conditions to guarantee the global exof Project FONDAP 15110019 is acknowledged 2σ(k − 1) v(k) σ(k) = (2) v(k) > = 00 The support of Project FONDAP 15110019 is acknowledged σ(k − 1) v(k) = 0 The support of Project FONDAP 15110019 is acknowledged
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3. CLOSED-LOOP STABILITY In this section, as a first step, the necessary conditions to ensure stability for the switched system are obtained. The main idea is to find a CLKF for all modes using a common PI controller. Firstly the closed-loop equations are described and by using Lyapunov arguments the stability results are obtained. 3.1 Closed-loop model Let us consider a PI controller defined as: k−1 u(k) = kp e(k) + ki e(i)
(3)
i=0
where kp ∈ R and ki ∈ R are the parameters to be tuned.
The error signal is defined as e(k) = r(k) − y(k), where r(k) ∈ R is the reference. For simplicity and without loss of generality we consider r(k) = 0 for all k.
The augmented state vector xa (k) ∈ Rn+1 and output vector ya (k) ∈ R2 are defined as: x(k) y(k) k−1 k−1 , y (k) = (4) xa (k) = y(i) a y(i) i=0
i=0
The new augmented system is defined as xa (k + 1) = Aσ xa (k) + bσ u(k − dσ(k) )
(5)
ya (k) = C xa (k)
where
Aσ 0 bσ c 0 , bσ = ,C = c 1 0 01
Aσ =
Thus, the PI controller can be written as an output control law: (6) u(k) = −kc ya (k) = −kc Cxa (k) where (7) kc = (kp ki ) In order to write the closed-loop equations with the same number of states for both modes, a simple approach is considered. Let us consider d1 ≥ d2 and h defined as (8) h = d1 − d 2 . By defining a new state vector z(k) with appropriate dimension xa (k) xa (k − 1) , (9) z(k) = .. .
xa (k − d2 ) a new representation of (5) can be written as z(k + 1) = fσ (z(k))
(10)
where
with
A1 =
1 z(k) + Ap (z(k) − z(k − h)) f1 (k) = A 2 z(k) f2 (k) = A
A1 0 · · · 0 −b1 kc C I 0 ··· 0 0 0 I ··· 0 0
. . . 0
. . . . . . . . . 0 ··· I
. . . 0
A2 0 · · · 0 −b2 kc C I 0 ··· 0 0 0 I ··· 0 0
, A 2 =
. . . 0
. . . . . . . . . 0 ··· I
. . . 0
(11)
Ap =
0 0 · · · 0 b1 kc C 0 0 ··· 0 0 0 0 ··· 0 0
..
. . . . .. . . . . 0 0 ··· 0
. . . 0
125
3.2 Lyapunov stability 2 is a Schur First, let us assume that kc is such that A matrix, i.e. all its eigenvalues have magnitude less than one. In order to establish the stability of the closed loop system (10), we propose a common candidate LyapunovKrasovskii function defined as in [Chen et al. 2003]; i.e. V (k) = V1 (k) + V2 (k) + V3 (k) (12) where V1 (k) = zT (k)P1 z(k) 0 k−1 V2 (k) = wT (l)S1 w(l) (13) θ=−h+1 l=k−1+θ k−1 V3 (k) = zT (l)S2 z(l) l=k−h
with
w(k) = z(k + 1) − z(k) (14) T P1 = P1 > 0 (15) S1 = ST1 > 0 (16) S2 = ST2 > 0 (17) The difference of the candidate function, ∆V (k), is defined as follows ∆V (k) = V (k + 1) − V (k) (18) = ∆V1 (k) + ∆V2 (k) + ∆V3 (k) The conditions to ensure ∆V (k) ≤ 0 are obtained for each mode. 3.3 Conditions for σ = 1 Analyzing each term ∆V1 (k) = 2zT (k)P1 w(k) + wT (k)P1 w(k) ∆V2 (k) = wT (k)hS1 w(k) −
k−1
wT (l)S1 w(l)
l=k−h
∆V3 (k) = zT (k)S2 z(k) − zT (k − h)S2 z(k − h) The first term of ∆V1 (k) can be bounded above with the Moon’s Inequality, as in [Chen et al. 2003], T T T w(k) 2z (k)P1 w(k) = 2η (k)P (19) 0 z(k) P1 0 where η(k) = and P = , with P2 and P3 w(k) P 2 P3 arbitrary matrices of appropriate dimensions. From the equation (11) with σ = 1, it follows k−1 1 − I)z(k) − w(k) + Ap 0 =(A w(l) (20) l=k−h
Therefore, we can write the equation (19) as w(k) w(k) 2η T (k)PT = 2η T (k)PT 1 − I)z(k) − w(k) 0 (A k−1
T
− 2η (k)P
405
T
l=k−h
0 p w(l) A
(21)
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p = −Ap . The second term of the equation (21) where A can be bounded above by using the Moon’s Inequality, as follows k−1 0 η T (k)PT −2 p w(l) A l=k−h k−1 0 T W M − PT η(k) p η(k) A ≤ w(l) w(l) Tp P l=k−h MT − 0 A S1 = hη T (k)Wη(k) + 2η T (k) M − PT
+
k−1
0 Ad
(z(k) − z(k − h))
T
w (l)S1 w(l)
T P1 A 2 − P1 z(k) ∆V1 (k) = zT (k) A 2
∆V2 (k) = wT (k)hS1 w(k) − T
l=k−h
T
∆V3 (k) = z (k)S2 z(k) − z (k − h)S2 z(k − h) 2 is a Schur matrix, it is possible to In this case, since A prove that ∆V2 (k) < 0 and ∆V3 (k) < 0. For instance, let us consider ∆V3 (k) which can be written in terms of the delayed variable z(k − h) as follows ∆V3 (k) = zT (k)S2 z(k) − zT (k − h)S2 z(k − h) T )h S2 (A 2 )h z(k − h) = zT (k − h)(A − zT (k − h)S2 z(k − h)
with
W M ≥0 MT S1 W1 W2 W= W T W3 2 M1 M= M2 Finally, equation (18) can be bounded above as
(22)
2 is a Schur matrix, then If A T h 2 )h < S2 (A ) S2 (A 2
(23)
Thus, the conditions are reduced to the following LMI 2 − P1 < 0 T P1 A (28) A 2
Summarizing, the stability of the closed-loop system is stablished if there exist matrices P1 , P2 ,P3 ,S1 ,S2 ,W1 ,W2 , W3 ,M1 and M2 such that LMIs (15),(16),(17),(22),(26), (28) are satisfied.
+ wT (k) [P1 + hS1 ] w(k) w(k) + 2η T (k)PT 1 − I z(k) − w(k) A
4. CONTROLLER TUNING In this section we consider the following performance criteria in order to tune the PI controller.
− zT (k − h)S2 z(k − h)
which can be expressed as
J= T
∆V (k) ≤ z(k) (k)Λz(k) where the new state vector z(k) is defined as z(k) η(k) y(k) z(k) = = z(k − h) z(k − h) and matrix Λ as 0 T − M Ψ P p A Λ = with T P − MT −S2 0A p T S2 0 M Ψ = hW + + [M 0] + 0 P1 + hS1 0
(24)
(25)
∞
zT (i)Qz(i)
(29)
i=0
with Q > 0.
If the in-equations (26) and (28) are bounded by −Q; i.e. Λ < −Q 2 − P1 < −Q T P1 A A
(30)
(31) then the difference of the candidate Lyapunov function (18) can be bounded as follows 2
∆V (k) = V (k + 1) − V (k) ≤ −zT (k)Qz(k)
(32)
By adding both sides of (32) from k = 0 to k = ∞, a bound for J can be obtained ∞ ∞
T − A T − I 0 I 0A 1 p P + A1 − Ap − I −I I −I
i=0
Therefore, if
Λ<0
(27)
and therefore ∆V3 (k) < 0. Similar procedure can be used to establish ∆V2 (k) < 0.
∆V (k) ≤ hη T (k)Wη(k) + zT (k)S2 z(k) 0 (z(k) − z(k − h)) + 2η T (k) M − PT Ap
+PT
wT (l)S1 w(l)
2
l=k−h
k−1
(26)
then ∆V (k) ≤ 0.
[V (i + 1) − V (i)] ≤ −
zT (i)Qz(i)
i=0
⇔ V (∞) − V (0) ≤ −J
As the closed-loop system is asymptotically stablem then V (z(∞)) = 0. Thus, the cost function is bounded by J ≤ V (0) = zT (0)(P1 + hS2 )z(0)
3.4 Conditions for σ = 2
(33)
Therefore, the minimization of (29) is equivalent to the minimization of the maximum eigenvalue of P1 + hS2 .
Analyzing each term 406
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127
of its evolution. When y(k) decreases the phenomenon influencing turbidity is sedimentation; but if y(k) increases the phenomenon is dominated by diffusion. The delay associated to the sedimentation process is lower than the delay related to the diffusion process, because the last one will depend on the structure of the plant. In fact, the diffusion delay will depend on the distance between the sensor location and the flow inlet pipe. Table 1. Parameters of the plant σ=1 0.90 0.02 3 1.00
Aσ bσ dσ c
σ=2 0.50 0.08 1 1.00
Table 1 shows the parameters used to the first simulation. Notice the different time delays associated to each mode.
The controller tuning problem can be stated as follows: find the parameters kp and ki so that the closed loop system with system (1) and the PI discrete controller, (3), is asymptotically stable and minimize the maximum eigenvalue of P1 + hS2 . Following similar ideas as in [da Silva Jr. et al. 2014], for a given set of controller parameters the matrix inequalities (26) and (28) are linear in the variables. Thus, these conditions can be used to certify a certain guarantee cost performance with respect to (29). Since the number of parameter is small, the tuning procedure will consider a grid on the controller parameters and iteratively will check 2 is Schur and then will solve the following first that A optimization problem: trace(P1 + hS2 ) min P1 ,P2 ,P3 ,S1 ,S2 ,W1 ,W2 ,W3 ,M1 ,M2 (34) s.t.(15), (16), (17), (22), (30), (31) for each point of the grid.
Considering a grid of parameters, the region D1 , where the closed-loop systems associated to mode 2 is stable; i.e. 2 ) ) < 1 D1 = (kp , ki ) : max( λ(A (35) is obtained.
The LMI feasibility region D2 is defined as the region where exist matrices P1 , P2 ,P3 , S1 ,S2 ,W1 ,W2 ,W3 ,M1 and M2 such that (15),(16),(17),(22),(26),(28) are satisfied. Notice that D2 ⊆ D1 . This region is depicted in Figure 2 along with D1 . 7
6
5
4
3
5. EXAMPLE
2
The example describes an application of the proposed method to a sedimentation process. Sedimentation systems are very important in water treatment plants. Sedimentation columns, as depicted in Figure 1, are equipments used to separate solids from residual water. The complex dynamics of this process can be described by nonlinear partial differential equations [Concha 2014]. However, a simple direction dependent empirical model can be obtained by using measurements of the volumetric flow rates and concentration at a certain point of the sedimentation column. The solid concentration is measured by a turbidity meter at the top of the column.
1
The main variables associated to the empirical model are: f1 (k): input flow of residual water f2 (k): output flow with high concentration of minerals u(k) : clean water overflow y(k) : turbidity measurement
2 is Schur A LMI kc with Q1 kc with Q2 kc with Q3
ki
Fig. 1. Sedimentation Column.
0 -5
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Fig. 2. Search region, Stable regions for different kc and optimum values for different Q. Finally, the solution of the optimization problem (34), (kp∗ , ki∗ ), is also depicted in Figure 2. The optimal values for different parameters of Q are summarized in table 2. The temporal responses of the closed-loop system for different values of Q and arbitrary initial condition are shown in Figures 3 and 4 for fall and rise behavior of y(k) respectively. Table 2. Optimum kc
The plant can be modeled by equation (1), where the values of σ(k) depend of the direction of y(k). Different phenomena drives the turbidity depending on the direction
20
kp
Q1 Q2 Q3
Q 0.01 1.00 100
kp 3.8101 7.3418 9.3038
ki 1.0684 1.6975 1.5177
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4
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Now, we will compare this example with the worst case where we just consider one mode of the plant to tune the controller, i.e. the tuning procedure just consider the mode σ1 . Table 3 shows the parameter used to tune the new controller and the real parameters of the plant used in the following simulation. Note that, the real parameters of table 3 are the same used before in table 1. Table 3. Parameters to tuning the worst case. Aσ bσ dσ c
ki 0.6785 1.2494 1.1975
0
Fig. 4. Closed loop rise simulations for different Q
σ = 2(tuning) 0.90 0.02 3 1.00
kp 21.9620 18.9241 22.3418
y1 (k) y3 (k)
100
tiempo [s]
σ=1 0.90 0.02 3 1.00
Q 0.01 1.00 100
initial conditions. Even if these temporal responses seem to be a stable system, the stability of the closed loop system can not be ensure, because in this example the overall system have not been considered in the tuning process.
2
1
25
Table 4. Optimum kc
20
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Fig. 5. Search region, Stable regions for different kc and optimum values for different Q for the worst case.
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tiempo [s]
σ = 2(real) 0.50 0.08 1 1.00
Fig. 6. Closed loop simulations for different Q in the worst case
For this new scenario, the stability of the closed loop system can not be ensured. Figure 5 shows the region and optimum values for this ˜ 2 is a Schur new case. Note that, the region where A matrix is the same of the valid region for the LMI, i.e. D2 = D1 . The optimal values for different parameters of Q are summarized in table 4. Figure 6 shows a temporal response of the switched systems in closed loop with inadequate tuning. It is clear that the overall system is unstable for Q1 and Q3 . The temporal responses for Q2 are shown in figures 7 and 8 for different 408
6. CONCLUSIONS Direction dependent models can represent a wide range of nonlinear process. In this work, a simple methodology for tuning PI controllers for direction dependent systems with time delay has been outlined. The parameters are obtained by solving an optimization problem considering a quadratic performance criteria and a common quadratic Lyapunov-Krasovskii function. The extension to the Multiple-Input Multiple-Output case can be achieved in a similar way if all the inputs have the same delay dσ . The results obtained by using an a simple direction dependent model of a sedimentation column are encouraging.
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Concha, F. (2014). Solid-Liquid separation in the mining industry. Springer. da Silva Jr., J.G., Lages, W., and Sbarbaro, D. (2014). Event-triggered pi controler design. In Proceedings of the 19th IFAC World Congress, 6947–6952. Leith, D., Shorten, R., and Leithead, W. (2003). Issues in the design of switched linear control system: A benchmark study. International Journal of Adaptative Control and Signal Processing, 17(2). Lin, H. and Antsaklis, P. (2009). Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Transactions on Automatic Control, 54(2), 308– 325. Tan, A. (2009). Direction-dependent systems - a survey. Automatica, 45(12), 2729–2743.
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Fig. 7. Closed loop simulations for Q2 in the worst case 2
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Fig. 8. Closed loop simulations for Q2 in the worst case Further work considering a real time implementation of this strategy is underway. In addition, additional research will consider different performance criteria, switched Lyapunov functions and the design of a direction dependent PI controller. REFERENCES Chaib, S., Boutat, D., Benali, A., and Kratz, F. (2006). Dynamic controller of switched linear systems : a common lyapunov function approach. In Proceedings of the 45th IEEE Conference on Decision and Control, 125– 130. Chen, J., Wu, I., lien, C., lee, C., Chen, R., and Yu, K. (2014). Robust exponential stability for uncertain discrete-time switched systems with interval timevarying delay through a switching signal. Journal of Applied Research and Technology, 12(6), 1187–1197. Chen, W., Guan, Z., and Lu, X. (2005). Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach. Systems & Control Letters, 54, 547–555. Chen, W., Guan, Z., and X.Lu (2003). Delay-dependent guaranteed cost control for uncertain discrete-time system with delay. EIT Proceedings: Control Theory and Application, 27(3). 409
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IFAC PapersOnLine 51-26 (2018) 124–129
Tuning discrete PI controllers Tuning discrete PI controllers Tuning discrete PI controllers direction-dependent systems with Tuning discrete PI controllers direction-dependent systems with direction-dependent systems direction-dependent systems with with J. Toledo ∗∗ D. Sbarbaro ∗∗
for for for delay for delay delay delay
J. Toledo ∗∗∗ D. Sbarbaro ∗∗∗ J. Toledo ∗ D. Sbarbaro ∗ ∗ J. Toledo D. Sbarbaro of Electrical o ∗ ∗ Department of Electrical Engineering, Engineering, Universidad Universidad de de Concepci´ Concepci´ on, n, ∗ Department ∗ Concepci´ o n, Chile Engineering, (e-mail: [email protected], Department of Electrical Universidad de Concepci´ on, ∗ Concepci´ o n, Chile (e-mail: [email protected], Department of Electrical Engineering, Universidad de Concepci´ on, [email protected]) Concepci´ on, Chile (e-mail: [email protected], [email protected]) Concepci´ on, Chile (e-mail: [email protected], [email protected]) [email protected]) Abstract: Abstract: Direction-dependent Direction-dependent models models can can represent represent aa wide wide range range of of nonlinear nonlinear systems. systems. The The design of a Direction-dependent discrete time PI controller for these type ofa systems withoftime delay systems. is addressed. Abstract: models can represent wide range nonlinear The design of a discrete time PI controller for these type of systems with time delay is addressed. Abstract: models can represent wide range oftime nonlinear systems. The The asymptotic stability of the closed loop system is established by means ofdelay Lyapunov theory. design of a Direction-dependent discrete time of PIthe controller for these type ofa systems with is addressed. The asymptotic stability closed loop system is established by means Lyapunov theory. design of a discrete time PIthe controller for these type of systems with timeof delay is addressed. tuning strategy is based on solving an optimization problem with LMI constraints. A The asymptotic stability of closed loop system is established by means of Lyapunov theory. The asymptotic tuning strategy is based on solving an optimization problem with of LMI constraints. A The stability of the the on closed loop an system isproposed established by means Lyapunov theory. simple example illustrates effectiveness of the approach. Extensions to include The tuning strategy is based solving optimization problem with LMI constraints. A simple example illustrates the effectiveness of the proposed approach. Extensions to include The strategy is based solving anofoptimization problem withExtensions LMI are constraints. A more tuning complex temporal closed-loop specifications and switched PI controllers part of our simple example illustrates the on effectiveness theand proposed approach. to include more complex temporal closed-loop specifications switched PI controllers are part of our simple example illustrates the effectiveness of the proposed approach. Extensions to include current interest. more complex closed-loop specifications and switched PI controllers are part of our current interest.temporal more complex current interest.temporal closed-loop specifications and switched PI controllers are part of our © 2018, IFAC (International Federation of Automatic Control) Hosting by ElsevierInequality. Ltd. All rights reserved. current interest. Keywords: Keywords: PID PID controller, controller, direction-dependent direction-dependent dynamics, dynamics, Linear Linear Matrix Matrix Inequality. Keywords: PID controller, direction-dependent dynamics, Linear Matrix Inequality. Keywords: PID controller, direction-dependent dynamics, Linear Matrix Inequality. 1. INTRODUCTION ponential stability of the closed loop system are obtained. obtained. 1. INTRODUCTION ponential stability of the closed be loop systemtoare These results can directly applied the class 1. INTRODUCTION ponential stability of the closed be loop systemtoare These results can not not directly applied theobtained. class of of 1. INTRODUCTION ponential stability of the closed loop system are obtained. systems addressed in this work, since the system time delay Direction-dependent systems are non linear system such These results can not directly be applied to the class of addressed in this work, since the system time delay Direction-dependent systems are non linear system such systems results can in not directly betoapplied tothis the class of is direction dependent. In order address problem, that their dynamic switches different Direction-dependent systemsbetween are nontwo linear systemmodes such These systems addressed this work, since the system time delay is direction dependent. Inwork, ordersince to address thistime problem, that their dynamic switches between two different modes systems addressed in this the system delay Direction-dependent systems are non linear system such in this work, a simple transformation is proposed in order depending on the direction of the input or outputmodes vari- in direction In order to address this problem, that their dynamic switches of between two different this work,dependent. a simple transformation is proposed in order depending on theofdirection the input or outputmodes vari- is is direction InLyapunov-Krasovskii order to address thiscandidate problem, that switches can between two different to define aa dependent. CLKF. ables.their Thisdynamic kind systems be input found in many indusin this work, a simpleThe transformation is proposed in order depending on the direction of the or output varito define CLKF. The Lyapunov-Krasovskii candidate ables. This kind of systems can be found in many indusin this work, a simple transformation is proposed in et order depending on the direction of the input or output vari- function isa similar to theLyapunov-Krasovskii one proposed in [Chen al. trial processes such as sedimentation, heating or cooling, to define CLKF. The candidate ables. This kind of systems can be found in many industrial processes such as sedimentation, heating or cooling, function is similar to the one proposed in [Chen et al. to define a CLKF. The Lyapunov-Krasovskii candidate ables. This kind of systems can be found in many indus2003] and [Chen et al. 2005]. chemical, among others. Tan [2009] provides a unified view function is similar to the one proposed in [Chen et al. trial processes such as sedimentation, heating or cooling, 2003] and [Chen et al. 2005]. chemical, among others. Tan [2009] provides a unified view is[Chen similar to 2005]. the one proposed in [Chen et al. trial processes such as systems. sedimentation, heating or cooling, of direction-dependent These type ofa unified systems can function 2003] and et al. chemical, among others. Tan [2009] provides view is as of direction-dependent systems. These type of systems can The 2003]paper and [Chen et al. 2005]. chemical, among Tan provides view paper is organized organized as follow. follow. Sections Sections 22 describes describes be direction-dependent regarded as a others. special type[2009] ofThese linear switching systems; of systems. type ofa unified systems can The the characteristics of the direction-dependent In be regarded as a special type of linear switching systems; The paper is organized as follow. Sections 2systems. describes of direction-dependent systems. These type of systems can the characteristics of the direction-dependent systems. In where the switching function is defined by the process be regarded as a special type of linear switching systems; The paper is organized asmodels follow.are Sections 2systems. describes where the switching function islinear defined by thesystems; process section 3, the closed-loop described and the the characteristics of the direction-dependent In be regarded as a special type of switching 3, the closed-loop models are described and the itself. the switching function is defined by the process section where characteristics offor theclosed-loop direction-dependent systems. In necessary stability are obtained. section 3, conditions the closed-loop models are described and the itself. the switching function is defined by the process the where necessary conditions for closed-loop stability are obtained. itself. section 3,describes the closed-loop models to are described and the Section 4 a methodology tune a PI controller. The stability of linear switching system not only depends necessary conditions for closed-loop stability are obtained. itself. The stability of linear switching system not only depends Section a for methodology tostability tune a PI controller. necessary conditions closed-loop are obtained. Sections44 5describes illustrates the main results means of an on the estructure andswitching the parameters of only the different describes a methodology to tuneby a PI controller. The stability of linear system not depends Section Sections 5 illustrates the main results by means of an on the estructure and the parameters of the different Section 4 describes a methodology to tune a PI controller. The stability of linear switching system not only depends example. Finally in section 6 some conclusions and future modes, but also onand thethe switching function. Thedifferent neces- Sections 5 illustrates the main results by means of an on the estructure parameters of the example. Finally in section 6 some conclusions and future modes, but also on the switching function. The necesSections 5outlined. illustrates the main by means of an on estructure and the parameters of the different are sarythe andbut sufficient conditions for asymptotic stability of work example. Finally in section 6 someresults conclusions and future modes, also on the switching function. The neceswork are outlined. sary andbut sufficient for are asymptotic stability of example. Finally in section 6 some conclusions and future modes, also onconditions theswitching switching function. The necessystems with arbitrary surveyed in Lin and sary and sufficient conditions for asymptotic stability of work are outlined. systems with arbitrary switching are surveyed stability in Lin and sary andwith sufficient conditions asymptotic of work are 2. outlined. DIRECTION-DEPENDENT SYSTEMS Antsaklis [2009]. One ofswitching the keyfor assumption is that all the systems arbitrary are surveyed in Lin 2. DIRECTION-DEPENDENT SYSTEMS Antsaklis [2009]. One of the key assumption is that all and the systems with arbitrary switching are surveyed in Lin and 2. DIRECTION-DEPENDENT SYSTEMS modes are described by models of the same order. To the Antsaklis [2009]. Oneby of models the key of assumption that To all the the modes are described the same is order. 2. DIRECTION-DEPENDENT SYSTEMS Antsaklis [2009]. Oneby of the the key assumption is that To all the the In this work, best of our knowledge, study of direction-dependent modes are described models of the same order. In this work, we we will will consider consider SISO SISO systems systems with with directiondirectionbest of are ourdescribed knowledge, the study of direction-dependent modes by models of the same order. To the dependent dynamics which can be systems written as: systems with direction dependent time delays has not been In this work, we will consider SISO with directionbest of our knowledge, the study of direction-dependent dependent dynamics which can be written as: systems with direction dependent time delays has not been In this work, consider SISO systems with best of our knowledge, the study of direction-dependent addressed. The discrete models associated to these systems +be bσ(k) u(k −as: dσ(k)direction) x(kdynamics +we 1)will = A which can written systems with direction time delays has not been dependent σ(k) x(k) addressed. The discretedependent models modes associated to these systems bσ(k) u(k −as: dσ(k) x(kdynamics + 1) = Awhich dependent can+be written σ(k) (1) systems with direction dependent time delays has not been σ(k) x(k) σ(k) σ(k) ) lead to representations having described by models = (1) addressed. The discrete models associated to these systems x(k) + b u(k − d ) x(k + 1) A σ(k) σ(k) σ(k) y(k) = c x(k) σ(k) lead to representations described bysystems models addressed. Theorders. discrete having models modes associated to these y(k) x(k) + bσ(k) u(k − dσ(k) ) x(k + 1) = c Ax(k) (1) with different σ(k) σ(k) σ(k) lead to representations having modes described by models n = c x(k) with different orders. y(k) (1) x(k) ∈ R is the state vector, u(k) ∈ R is the input, n lead to representations having modes described by models where n= with different orders. where x(k) ∈ R is the state vector, u(k) ∈ R is the input, y(k) c x(k) n output and σ(k) is a discrete variable This work addresses the problem of tuning a PI controller where n y(k) ∈x(k) R is∈ the with different orders. R is the state vector, u(k) ∈ R is the input, This worktype addresses the problem of tuning a PIiscontroller y(k) ∈x(k) R is the andvector, σ(k)simplicity is a discrete variable where Rn isoutput the signal. state u(k) ∈ Rwe is the input, for of The analysis centered This worktype addresses the problem of tuning a PIiscontroller will write y(k) ∈ R is∈switching the output and For σ(k)simplicity is a discrete variable for these these of systems. systems. The stability stability analysis centered representing representing switching signal. For we will write n×n This work addresses the problem of tuning a PI controller y(k) ∈ R is the output and σ(k) is a discrete variable in finding a Common Quadratic Lyapunov Function for σ(k) for these type of systems. The stability analysis is centered as σ. The parameter of the plant are A ∈ R n×n representing switching signal. For simplicity weσσ will write n×n ,, σ(k) as σ. The parameter of the plant are A ∈ R in finding a Common Quadratic Lyapunov Function for n×1 1×n for these type of systems. The stability analysis is centered σ n×n representing switching signal. For simplicity we will write establishing the closed Quadratic loop stability [Leith Function et al. 2003]. n×n b ∈ R , c ∈ R and d ∈ N. Notice also that σ(k) as σ. The parameter of the plant are A ∈ R in finding a Common Lyapunov for n×1 1×n σ σ σ also n×n n×1 , c ∈ R1×n and d σ bσσ ∈asRσ. ∈ N. are Notice that,, establishing the closed loop stability [Leith et design al. 2003]. σ in finding a Common Quadratic Lyapunov Function for σ n×1 1×n σ(k) The parameter of the plant A ∈ R The use of a CQLF in the context of control for n×1 1×n σ also the time delay depends on σ(k). The switching signal, establishing the closed loop stability [Leith et al. 2003]. b ∈ R , c ∈ R and d ∈ N. Notice that σ σ σ timen×1 the delay depends σ(k). switching The use of athe CQLF in loop the context of[Leith control design for b establishing closed stability etby al. several 2003]. R (2), , ccan ∈ depend R1×n on and dσσ on ∈The N. Notice alsosignal, switched linear systems have been addressed σ ∈ The use of a CQLF in the context of control design for equation either the switching direction ofthat the the time delay depends on σ(k). The signal, switched linear systems have been addressed by several (2), can depend either onThe the switching direction of the The use For of ainstance, CQLF in the context of control design for equation the time delay depends on σ(k). signal, authors. Chaib et been al. [2006] provideby a method input: v(k) = u(k) − u(k − 1) or the output: v(k) = y(k) − equation (2), can depend either on the direction of the switched linear systems have addressed several v(k) = u(k) u(k − 1) or the v(k) = y(k) − authors. For instance, Chaib etfor al.continuous [2006] provide aswitched method switched linear systems have been addressed several input: equation (2), can − depend either onoutput: the direction of the to designFor dynamic controllers timeby y(k − 1). It takes its values over a finte set of values input: v(k) = u(k) − u(k − 1) or the output: v(k) = y(k) − authors. instance, Chaib et al. [2006] provide a method y(k − 1). It takes its values over a finte set of values to design dynamic controllers for continuous time switched authors. For instance, etfor al.continuous [2006] provide aswitched method v(k) =the u(k)different −its u(kvalues − 1) orover theFor output: v(k) = y(k) − linear system. Chen etChaib al. [2014] address the problem of input: representing modes. instance σ(k) = to design dynamic controllers time y(k − 1). It takes a finte set of values the different modes. For instance σ(k) = ii linear system. Chen etsignal al. [2014] address the problem of representing to design dynamic controllers for continuous time switched y(k − 1). It takes its values over a finte set of values designing a switching for exponential stabilization linear system. Chen et al. [2014] address the problem of means that the ith mode, represented by A , b and d , i isi representing theith different modes. For by instance designing a switching signal for exponential stabilization means that the mode, represented Aiii , biii σ(k) and d= ii , isi linear system. Chen etsignal al. system [2014] address thestabilization problem of representing theith different modes. For by instance σ(k) = of a discrete-time switched with time-varying delay. activated. means that the mode, represented A , b and d designing a switching for exponential ii ii ii , is activated. of a discrete-time switched system with time-varying delay. designing a switching signal for exponential stabilization means that the ith mode, represented by A , b and d Bya applying a Common Lyapunov-Krasovskii Functional i 0 i i , is 1 v(k) < activated. of discrete-time switched system with time-varying delay. By applying a Common Lyapunov-Krasovskii Functional 1 v(k) < 0 of a discrete-time switched system with time-varying delay. activated. (CLKF), sufficient conditions to guarantee the global ex2 v(k) > σ(k) (2) By applying a Common Lyapunov-Krasovskii v(k) (CLKF), sufficient conditions to guarantee theFunctional global exv(k) < > 0000 σ(k) = = 112 (2) By applying a Common Lyapunov-Krasovskii v(k) < (CLKF), sufficient conditions to15110019 guarantee theFunctional global exσ(k − 1) v(k) = 0 2 v(k) > 0 σ(k) = (2) The support of Project FONDAP is acknowledged σ(k − 1) v(k) = 0 The support (CLKF), sufficient conditions to guarantee the global exof Project FONDAP 15110019 is acknowledged 2σ(k − 1) v(k) σ(k) = (2) v(k) > = 00 The support of Project FONDAP 15110019 is acknowledged σ(k − 1) v(k) = 0 The support of Project FONDAP 15110019 is acknowledged
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J. Toledo et al. / IFAC PapersOnLine 51-26 (2018) 124–129
3. CLOSED-LOOP STABILITY In this section, as a first step, the necessary conditions to ensure stability for the switched system are obtained. The main idea is to find a CLKF for all modes using a common PI controller. Firstly the closed-loop equations are described and by using Lyapunov arguments the stability results are obtained. 3.1 Closed-loop model Let us consider a PI controller defined as: k−1 u(k) = kp e(k) + ki e(i)
(3)
i=0
where kp ∈ R and ki ∈ R are the parameters to be tuned.
The error signal is defined as e(k) = r(k) − y(k), where r(k) ∈ R is the reference. For simplicity and without loss of generality we consider r(k) = 0 for all k.
The augmented state vector xa (k) ∈ Rn+1 and output vector ya (k) ∈ R2 are defined as: x(k) y(k) k−1 k−1 , y (k) = (4) xa (k) = y(i) a y(i) i=0
i=0
The new augmented system is defined as xa (k + 1) = Aσ xa (k) + bσ u(k − dσ(k) )
(5)
ya (k) = C xa (k)
where
Aσ 0 bσ c 0 , bσ = ,C = c 1 0 01
Aσ =
Thus, the PI controller can be written as an output control law: (6) u(k) = −kc ya (k) = −kc Cxa (k) where (7) kc = (kp ki ) In order to write the closed-loop equations with the same number of states for both modes, a simple approach is considered. Let us consider d1 ≥ d2 and h defined as (8) h = d1 − d 2 . By defining a new state vector z(k) with appropriate dimension xa (k) xa (k − 1) , (9) z(k) = .. .
xa (k − d2 ) a new representation of (5) can be written as z(k + 1) = fσ (z(k))
(10)
where
with
A1 =
1 z(k) + Ap (z(k) − z(k − h)) f1 (k) = A 2 z(k) f2 (k) = A
A1 0 · · · 0 −b1 kc C I 0 ··· 0 0 0 I ··· 0 0
. . . 0
. . . . . . . . . 0 ··· I
. . . 0
A2 0 · · · 0 −b2 kc C I 0 ··· 0 0 0 I ··· 0 0
, A 2 =
. . . 0
. . . . . . . . . 0 ··· I
. . . 0
(11)
Ap =
0 0 · · · 0 b1 kc C 0 0 ··· 0 0 0 0 ··· 0 0
..
. . . . .. . . . . 0 0 ··· 0
. . . 0
125
3.2 Lyapunov stability 2 is a Schur First, let us assume that kc is such that A matrix, i.e. all its eigenvalues have magnitude less than one. In order to establish the stability of the closed loop system (10), we propose a common candidate LyapunovKrasovskii function defined as in [Chen et al. 2003]; i.e. V (k) = V1 (k) + V2 (k) + V3 (k) (12) where V1 (k) = zT (k)P1 z(k) 0 k−1 V2 (k) = wT (l)S1 w(l) (13) θ=−h+1 l=k−1+θ k−1 V3 (k) = zT (l)S2 z(l) l=k−h
with
w(k) = z(k + 1) − z(k) (14) T P1 = P1 > 0 (15) S1 = ST1 > 0 (16) S2 = ST2 > 0 (17) The difference of the candidate function, ∆V (k), is defined as follows ∆V (k) = V (k + 1) − V (k) (18) = ∆V1 (k) + ∆V2 (k) + ∆V3 (k) The conditions to ensure ∆V (k) ≤ 0 are obtained for each mode. 3.3 Conditions for σ = 1 Analyzing each term ∆V1 (k) = 2zT (k)P1 w(k) + wT (k)P1 w(k) ∆V2 (k) = wT (k)hS1 w(k) −
k−1
wT (l)S1 w(l)
l=k−h
∆V3 (k) = zT (k)S2 z(k) − zT (k − h)S2 z(k − h) The first term of ∆V1 (k) can be bounded above with the Moon’s Inequality, as in [Chen et al. 2003], T T T w(k) 2z (k)P1 w(k) = 2η (k)P (19) 0 z(k) P1 0 where η(k) = and P = , with P2 and P3 w(k) P 2 P3 arbitrary matrices of appropriate dimensions. From the equation (11) with σ = 1, it follows k−1 1 − I)z(k) − w(k) + Ap 0 =(A w(l) (20) l=k−h
Therefore, we can write the equation (19) as w(k) w(k) 2η T (k)PT = 2η T (k)PT 1 − I)z(k) − w(k) 0 (A k−1
T
− 2η (k)P
405
T
l=k−h
0 p w(l) A
(21)
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p = −Ap . The second term of the equation (21) where A can be bounded above by using the Moon’s Inequality, as follows k−1 0 η T (k)PT −2 p w(l) A l=k−h k−1 0 T W M − PT η(k) p η(k) A ≤ w(l) w(l) Tp P l=k−h MT − 0 A S1 = hη T (k)Wη(k) + 2η T (k) M − PT
+
k−1
0 Ad
(z(k) − z(k − h))
T
w (l)S1 w(l)
T P1 A 2 − P1 z(k) ∆V1 (k) = zT (k) A 2
∆V2 (k) = wT (k)hS1 w(k) − T
l=k−h
T
∆V3 (k) = z (k)S2 z(k) − z (k − h)S2 z(k − h) 2 is a Schur matrix, it is possible to In this case, since A prove that ∆V2 (k) < 0 and ∆V3 (k) < 0. For instance, let us consider ∆V3 (k) which can be written in terms of the delayed variable z(k − h) as follows ∆V3 (k) = zT (k)S2 z(k) − zT (k − h)S2 z(k − h) T )h S2 (A 2 )h z(k − h) = zT (k − h)(A − zT (k − h)S2 z(k − h)
with
W M ≥0 MT S1 W1 W2 W= W T W3 2 M1 M= M2 Finally, equation (18) can be bounded above as
(22)
2 is a Schur matrix, then If A T h 2 )h < S2 (A ) S2 (A 2
(23)
Thus, the conditions are reduced to the following LMI 2 − P1 < 0 T P1 A (28) A 2
Summarizing, the stability of the closed-loop system is stablished if there exist matrices P1 , P2 ,P3 ,S1 ,S2 ,W1 ,W2 , W3 ,M1 and M2 such that LMIs (15),(16),(17),(22),(26), (28) are satisfied.
+ wT (k) [P1 + hS1 ] w(k) w(k) + 2η T (k)PT 1 − I z(k) − w(k) A
4. CONTROLLER TUNING In this section we consider the following performance criteria in order to tune the PI controller.
− zT (k − h)S2 z(k − h)
which can be expressed as
J= T
∆V (k) ≤ z(k) (k)Λz(k) where the new state vector z(k) is defined as z(k) η(k) y(k) z(k) = = z(k − h) z(k − h) and matrix Λ as 0 T − M Ψ P p A Λ = with T P − MT −S2 0A p T S2 0 M Ψ = hW + + [M 0] + 0 P1 + hS1 0
(24)
(25)
∞
zT (i)Qz(i)
(29)
i=0
with Q > 0.
If the in-equations (26) and (28) are bounded by −Q; i.e. Λ < −Q 2 − P1 < −Q T P1 A A
(30)
(31) then the difference of the candidate Lyapunov function (18) can be bounded as follows 2
∆V (k) = V (k + 1) − V (k) ≤ −zT (k)Qz(k)
(32)
By adding both sides of (32) from k = 0 to k = ∞, a bound for J can be obtained ∞ ∞
T − A T − I 0 I 0A 1 p P + A1 − Ap − I −I I −I
i=0
Therefore, if
Λ<0
(27)
and therefore ∆V3 (k) < 0. Similar procedure can be used to establish ∆V2 (k) < 0.
∆V (k) ≤ hη T (k)Wη(k) + zT (k)S2 z(k) 0 (z(k) − z(k − h)) + 2η T (k) M − PT Ap
+PT
wT (l)S1 w(l)
2
l=k−h
k−1
(26)
then ∆V (k) ≤ 0.
[V (i + 1) − V (i)] ≤ −
zT (i)Qz(i)
i=0
⇔ V (∞) − V (0) ≤ −J
As the closed-loop system is asymptotically stablem then V (z(∞)) = 0. Thus, the cost function is bounded by J ≤ V (0) = zT (0)(P1 + hS2 )z(0)
3.4 Conditions for σ = 2
(33)
Therefore, the minimization of (29) is equivalent to the minimization of the maximum eigenvalue of P1 + hS2 .
Analyzing each term 406
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127
of its evolution. When y(k) decreases the phenomenon influencing turbidity is sedimentation; but if y(k) increases the phenomenon is dominated by diffusion. The delay associated to the sedimentation process is lower than the delay related to the diffusion process, because the last one will depend on the structure of the plant. In fact, the diffusion delay will depend on the distance between the sensor location and the flow inlet pipe. Table 1. Parameters of the plant σ=1 0.90 0.02 3 1.00
Aσ bσ dσ c
σ=2 0.50 0.08 1 1.00
Table 1 shows the parameters used to the first simulation. Notice the different time delays associated to each mode.
The controller tuning problem can be stated as follows: find the parameters kp and ki so that the closed loop system with system (1) and the PI discrete controller, (3), is asymptotically stable and minimize the maximum eigenvalue of P1 + hS2 . Following similar ideas as in [da Silva Jr. et al. 2014], for a given set of controller parameters the matrix inequalities (26) and (28) are linear in the variables. Thus, these conditions can be used to certify a certain guarantee cost performance with respect to (29). Since the number of parameter is small, the tuning procedure will consider a grid on the controller parameters and iteratively will check 2 is Schur and then will solve the following first that A optimization problem: trace(P1 + hS2 ) min P1 ,P2 ,P3 ,S1 ,S2 ,W1 ,W2 ,W3 ,M1 ,M2 (34) s.t.(15), (16), (17), (22), (30), (31) for each point of the grid.
Considering a grid of parameters, the region D1 , where the closed-loop systems associated to mode 2 is stable; i.e. 2 ) ) < 1 D1 = (kp , ki ) : max( λ(A (35) is obtained.
The LMI feasibility region D2 is defined as the region where exist matrices P1 , P2 ,P3 , S1 ,S2 ,W1 ,W2 ,W3 ,M1 and M2 such that (15),(16),(17),(22),(26),(28) are satisfied. Notice that D2 ⊆ D1 . This region is depicted in Figure 2 along with D1 . 7
6
5
4
3
5. EXAMPLE
2
The example describes an application of the proposed method to a sedimentation process. Sedimentation systems are very important in water treatment plants. Sedimentation columns, as depicted in Figure 1, are equipments used to separate solids from residual water. The complex dynamics of this process can be described by nonlinear partial differential equations [Concha 2014]. However, a simple direction dependent empirical model can be obtained by using measurements of the volumetric flow rates and concentration at a certain point of the sedimentation column. The solid concentration is measured by a turbidity meter at the top of the column.
1
The main variables associated to the empirical model are: f1 (k): input flow of residual water f2 (k): output flow with high concentration of minerals u(k) : clean water overflow y(k) : turbidity measurement
2 is Schur A LMI kc with Q1 kc with Q2 kc with Q3
ki
Fig. 1. Sedimentation Column.
0 -5
0
5
10
15
407
25
Fig. 2. Search region, Stable regions for different kc and optimum values for different Q. Finally, the solution of the optimization problem (34), (kp∗ , ki∗ ), is also depicted in Figure 2. The optimal values for different parameters of Q are summarized in table 2. The temporal responses of the closed-loop system for different values of Q and arbitrary initial condition are shown in Figures 3 and 4 for fall and rise behavior of y(k) respectively. Table 2. Optimum kc
The plant can be modeled by equation (1), where the values of σ(k) depend of the direction of y(k). Different phenomena drives the turbidity depending on the direction
20
kp
Q1 Q2 Q3
Q 0.01 1.00 100
kp 3.8101 7.3418 9.3038
ki 1.0684 1.6975 1.5177
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2
y1 (k) y2 (k) y3 (k)
0 -2
0
10
20
30
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90
3
100 2.5
20
0
10
20
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90
ki
u1 (k) u2 (k) u3 (k)
0 -20
2 is Schur A LMI kc with Q1 kc with Q2 kc with Q3
3.5
2
1.5
100
2
1
1
σ1 (k) σ2 (k) σ3 (k)
1.5
0
10
20
30
40
50
60
70
80
90
0.5
100
0
tiempo [s]
-5
0
5
10
Fig. 3. Closed loop fall simulations for different Q
y1 (k) y2 (k) y3 (k)
0
0
10
20
30
40
50
60
70
80
90
Q1 Q2 Q3
100
u1 (k) u2 (k) u3 (k)
0
0
10
20
30
40
50
60
70
80
90
100
σ1 (k) σ2 (k) σ3 (k)
1.5
5000 0
10
20
30
40
50
60
70
80
90
-5000
Now, we will compare this example with the worst case where we just consider one mode of the plant to tune the controller, i.e. the tuning procedure just consider the mode σ1 . Table 3 shows the parameter used to tune the new controller and the real parameters of the plant used in the following simulation. Note that, the real parameters of table 3 are the same used before in table 1. Table 3. Parameters to tuning the worst case. Aσ bσ dσ c
ki 0.6785 1.2494 1.1975
0
Fig. 4. Closed loop rise simulations for different Q
σ = 2(tuning) 0.90 0.02 3 1.00
kp 21.9620 18.9241 22.3418
y1 (k) y3 (k)
100
tiempo [s]
σ=1 0.90 0.02 3 1.00
Q 0.01 1.00 100
initial conditions. Even if these temporal responses seem to be a stable system, the stability of the closed loop system can not be ensure, because in this example the overall system have not been considered in the tuning process.
2
1
25
Table 4. Optimum kc
20
-20
20
Fig. 5. Search region, Stable regions for different kc and optimum values for different Q for the worst case.
1
-1
15
kp
1
0
10
20
×10 5
30
40
50
60
70
80
90
100
40
50
60
70
80
90
100
50
60
70
80
90
100
u1 (k) u3 (k)
0 -1
0
10
20
30
2
σ1 (k) σ3 (k)
1.5 1
0
10
20
30
40
tiempo [s]
σ = 2(real) 0.50 0.08 1 1.00
Fig. 6. Closed loop simulations for different Q in the worst case
For this new scenario, the stability of the closed loop system can not be ensured. Figure 5 shows the region and optimum values for this ˜ 2 is a Schur new case. Note that, the region where A matrix is the same of the valid region for the LMI, i.e. D2 = D1 . The optimal values for different parameters of Q are summarized in table 4. Figure 6 shows a temporal response of the switched systems in closed loop with inadequate tuning. It is clear that the overall system is unstable for Q1 and Q3 . The temporal responses for Q2 are shown in figures 7 and 8 for different 408
6. CONCLUSIONS Direction dependent models can represent a wide range of nonlinear process. In this work, a simple methodology for tuning PI controllers for direction dependent systems with time delay has been outlined. The parameters are obtained by solving an optimization problem considering a quadratic performance criteria and a common quadratic Lyapunov-Krasovskii function. The extension to the Multiple-Input Multiple-Output case can be achieved in a similar way if all the inputs have the same delay dσ . The results obtained by using an a simple direction dependent model of a sedimentation column are encouraging.
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Concha, F. (2014). Solid-Liquid separation in the mining industry. Springer. da Silva Jr., J.G., Lages, W., and Sbarbaro, D. (2014). Event-triggered pi controler design. In Proceedings of the 19th IFAC World Congress, 6947–6952. Leith, D., Shorten, R., and Leithead, W. (2003). Issues in the design of switched linear control system: A benchmark study. International Journal of Adaptative Control and Signal Processing, 17(2). Lin, H. and Antsaklis, P. (2009). Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Transactions on Automatic Control, 54(2), 308– 325. Tan, A. (2009). Direction-dependent systems - a survey. Automatica, 45(12), 2729–2743.
2
y2 (k) 0 -2
0
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u2 (k) 0 -50
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σ2 (k) 1.5 1
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Fig. 7. Closed loop simulations for Q2 in the worst case 2
y2 (k) 0 -2
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u2 (k) 0 -50
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Fig. 8. Closed loop simulations for Q2 in the worst case Further work considering a real time implementation of this strategy is underway. In addition, additional research will consider different performance criteria, switched Lyapunov functions and the design of a direction dependent PI controller. REFERENCES Chaib, S., Boutat, D., Benali, A., and Kratz, F. (2006). Dynamic controller of switched linear systems : a common lyapunov function approach. In Proceedings of the 45th IEEE Conference on Decision and Control, 125– 130. Chen, J., Wu, I., lien, C., lee, C., Chen, R., and Yu, K. (2014). Robust exponential stability for uncertain discrete-time switched systems with interval timevarying delay through a switching signal. Journal of Applied Research and Technology, 12(6), 1187–1197. Chen, W., Guan, Z., and Lu, X. (2005). Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach. Systems & Control Letters, 54, 547–555. Chen, W., Guan, Z., and X.Lu (2003). Delay-dependent guaranteed cost control for uncertain discrete-time system with delay. EIT Proceedings: Control Theory and Application, 27(3). 409